Abstracts of Research Talks
Andreas Basse-O'Connor
Aarhus University, Denmark
On the semimartingale property of moving averages
Continuous time moving averages, as e.g. the fractional Brownian motion, the Ornstein-Uhlenbeck process and their generalizations, but also Brownian semistationary processes, have been used repeatedly in turbulence, finance and related fields. The present talk is concerned with the question - when is a moving average a semimartingale? Various filtrations will be considered. In particular, necessary and sufficient conditions are provided for a moving average to be a semimartingale when the driving process is a Brownian motion, a Lévy process or a Gaussian chaos process.
Martin Greiner
Siemens AG, Corporate Research & Technology, 80200 Munich, Germany
Modelling of turbulent wind fields beyond the engineering standard
The engineering standard for the modelling of turbulent wind fields is based on Gaussian processes. It is not able to describe the strongly non-Gaussian and intermittent character of small-scale turbulence. Multifractal processes are introduced on top of the engineering approach. This generalized model is able to describe real small-scale turbulence with good precision. By simulation with wind-turbine interaction models and from data, it is shown that such multifractal wind fields produce significant finger prints in power and load time series of wind turbines.
Adam Jakubowski
Nicolaus Copernicus University, Torun, Poland
Are fractional Brownian motions predictable?
Motivated by the Graversen-Rao decomposition theorem we introduce a device, called the local predictor, which extends the idea of the predictable compensator. The local predictor of a martingale (in particular: of a Brownian motion) trivially exists and equals 0. It is shown that a fractional Brownain motion with the Hurst index greater than 1/2 coincides with its local predictor while fractional Brownian motions with the Hurst index smaller than 1/2 do not admit
any local predictor. Other classes of processes of non-semimartingale type are also studied.
Claudia Klüppelberg
Technische Universität, Zentrum Mathematik, Lehrstuhl für Mathematische Statistik München, Germany
Generalized fractional Lévy processes with fractional Brownian motion limit
Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional kernel functions to the more general class of regularly varying functions with the corresponding fractional integration parameter. This requires an extension of the Riemann-Liouville fractional integrals, which leads to a new fractional analysis. We invoke then this analysis to define stochastic integrals with respect to a generalized fractional Lévy process and investigate some of their properties, in particular their second order structure. For the fractional compound Poisson process (also known as a special multiplicative Poisson shot noise process) with finite variance a functional central limit theorem to fractional Brownian motion is well-known. We extend this result to a general Lévy framework; i.e. we prove a functional central limit theorem for stochastic integrals of a generalized fractional Lévy process. As a specific example we present our result for an Ornstein-Uhlenbeck process driven by a time scaled generalised fractional Lévy process.
This is joint work with Muneya Matsui (Keio University).
Alexander Lindner
Institute for Mathematical Stochastics, TU Braunschweig, Germany
On the sample autocorrelation function of Lévy driven continuous time moving average processes
Let L= (L_t) be a two-sided Lévy process with expectation 0 and finite variance and let f be an L^2-function. We consider the continuous time moving average process X_t = \int f(t-s) \, dL_s. Based on the observations at discrete times t=1,2,3,..., we study the asymptotic behaviour of the sample mean and the sample autocovariances. A limit theorem for the sample autocorrelations is also obtained together with a Bartlett-type formula. This formula differs significantly from the corresponding discrete time moving average process case, since in the discrete time setting, the fourth moment of the driving noise sequence drops out, which in general is not the case for the continuous time setting. The results have applications for the estimation of the Hurst parameter of fractional Lévy noise.
The talk is based on joint work with Serge Cohen (in preparation).
Yulia Mishura
Department of Probability, Statistics and Actuarial Mathematics, Kyiv National Taras Shevchenko University, Ukraine
Imperfect efficient hedging on a financial market involving Brownian and fractional Brownian motion
We consider the financial market involving both standard and fractional Brownian motion with Hurst index H>1/2. Two different cases are considered: ½<H<3/4 and ¾<H<1. In the first case the market can be considered, in some case, as usual Black-Scholes-Merton market, therefore, we can construct an efficient hedging for vanilla options and consider the different rates of lending and borrowing. In the second case we consider the approximations of the stock prices, give the formulae for imperfect hedging and study the asymptotic behavior of <br />the prices.
Andreas Neuenkirch
TU Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund
A Milstein-type scheme without Lévy-area terms for SDES driven by fractional Brownian motion
Recently, several Taylor-type approximation schemes have been proposed for stochastic differential equations (SDEs) driven by a fractional Brownian motion with Hurst parameter H in (1/4; 1), see e.g. A.M. Davie, Differential equations driven by rough paths: an approach via discrete approximation, Appl. Math. Res. Express (2007), No. 2 and P. Friz, N. Victoir, Multidimensional stochastic processes seen as rough paths, Cambridge University Press, to appear. Since the distribution of the arising iterated integrals is known only in particular cases, the approximation of these integrals is required (as in the case of the classical Taylor schemes for SDEs driven by a standard Brownian motion).
In this talk, we will present a Milstein-type scheme that uses only increments of the driving fractional Brownian motion and show its convergence for H > 1/3. Moreover, we also discuss the exact rate of convergence and the asymptotic error distribution of this scheme.
Rimas Norvaisa
Institute of Mathematics and Informatics, Vilnius, Lithuania
Partial sum processes in p-variation form
Abstract only available as PDF.
Mikko Pakkanen
Department of Mathematics and Statistics, University of Helsinki, Finland
Stochastic integrals and conditional full support
Guasoni, Rasonyi, and Schachermayer have shown that the conditional full support (CFS) property implies absence of arbitrage and enables one to solve superreplication problems under small proportional transaction costs. We resent conditions that imply CFS for process Z := H + K \cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is continuous and of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.
Jan Pedersen
Aarhus University, Denmark
Martingale increment processes
When studying a Brownian motion with time parameter in \mathbb{R} (rather than \mathbb{R}_+) one quickly realises that this process is not a (local) martingale in any filtration. Instead, it turns out that it is a so called martingale increment process.
In this talk we define and study martingale increment processes. Topics considered include the behaviour at minus infinity, existence and properties of (generalised) quadratic variations and compensator, and stochastic integration with respect to martingale increment processes. Examples will be given as well.
Victor Perez-Abreu
Research Center for Mathematics CIMAT, Guanajuato, Mexico
On the infinite divisibility of power semicircle distributions
Power semicircle laws appear as the marginals of uniform distributions on spheres in high-dimensional Euclidean spaces. A review of some results is presented including a genesis and the so-called Poincaré's theorem. The moments of these distributions are related to the super Catalan numbers and their Cauchy transform are derived in terms of hypergeometric functions. Some members of this class of distributions play the role of the Gaussian distribution with respect to additive convolutions in non-commutative probability, such as the free, the (anti) monotone and the Boolean convolutions.
In this talk we disccuss the infinite divisibility of other members of the class of power semicircle distributions with respect to these convolutions.
Anthony Reveillac
Institut für Mathematik, Humboldt-Universität zu Berlin, Germany
Convergence of the wheighted quadratic variations of some fractional Brownian sheets
In this talk we present a central limit theorem for the weighted quadratic variations of the standard Brownian sheet. This result leads to the construction of an asymptotically normal estimator of the diffusion coefficient of some two-parameter processes. In addition, we study the asymptotic behavior of the quadratic variations of some fractional Brownian sheets. We finally compare our results with those obtain very recently in the fractional Brownian motion case by I. Nourdin, D. Nualart, C.A. Tudor and myself.
Albert Shiryaev
Steklov Mathematical Institute, Moscow, Russia
On the notion of randomness
By Kolmogorov "... we call RANDOM these phenomena where we cannot find a regularity allowing us to predict precisely their results. Generally speaking there is no ground to believe that a random phenomenon should posses any definite probability. Therefore, we should have distinguished between randomness proper (as absence of any regularity) and stochastic randomness (which is the subject of the probability theory)" <<"On logical foundation of probability theory", Fourth USSR-Japan Symposium, Proceedings, 1982, Lecture Notes in Math., vol. 1021)>>
In the present talk we intend to give a survey of the probabilistic and algorithmic results on the problem "What is randomness?"
Robert Stelzer
Institute for Advanced Study & Centre for Mathematical Sciences, Technische Universität München , Germany
Multivariate supOU processes and a stochastic volatility model with possible long memory
Multivariate supOU processes are defined using a Lévy basis on the real numbers times the set of square matrices with all eigenvalues having strictly negative real parts. We discuss the existence, the finiteness of moments, the second order structure and important path properties, noting that the peculiarities of the underlying matrices cause new phenomena and features compared to the known univariate case. In particular, we give precise conditions for the validity of an analogue to the stochastic differential equation satisfied by Ornstein-Uhlenbeck type processes, which has been conjectured in the univariate case by Barndorff-Nielsen [2001, Superposition of Ornstein-Uhlenbeck type processes, Theory Probab. Appl. 45, 175-194], but not yet been proven. Our results also imply conditions when supOU processes are compatible with semimartingale integration theory.
Using the general results, we define supOU processes on the positive semi-definite matrices, which we use as the "instantaneous covariance matrix" process in a stochastic volatility model. After analysing some properties of the resulting stochastic volatility models, we give some examples which show in particular that long range dependence effects may arise.
This talk is based on joint work with Ole Eiler Barndorff-Nielsen.
Steen Thorbjørnsen
Aarhus University, Denmark
General Concepts of Independence, Infinite Divisibility and Lévy Processes
In the talk I will describe the main notions of independence in classical and non-commutative probability. I will also give an overview of the associated concepts of convolution, infinite divisibility and Lévy Processes.
Esko Valkeila
Department of Mathematics and System Analysis, TKK, Helsinki, Finland
Riemann-Stieltjes integrals and fractional Brownian motion
We assume that the fractional Brownian motion has Hurst index H > 1/2. In the case of Brownian motion every square integrable random variable has an integral representation as a stochastic integral with respect to Brownian motion. In the talk I will give some results what kind of random variables can be represented as Riemann-Stieltjes integrals with respect to fractional Brownian motion or geometric fractional Brownian motion. The financial motivation for the results will also be discussed.
The talk is based on joint works with E. Azmoodeh (TKK) and Y. Mishura (Kiev), and E. Azmoodeh and H. Tikanmäki.
Almut Veraart
Aarhus University, Denmark
Modelling electricity spot and forward prices by ambit fields
This paper presents new models for electricity spot and forward prices, which are based on Lévy-driven semistationary processes and on ambit fields, respectively. Such models are able to capture many of the stylised facts observed in energy markets in a novel way. The main difference to the traditional models lies in the fact that we do not model the dynamics but the spot and the forward price directly. First, we give a detailed account on the probabilistic properties of the new models and we discuss asset pricing within the new model class. Also, we provide conditions under which the new models for the spot and the forward can be directly linked to each
other.
(This is joint work with F. E. Benth and O. E. Barndorff-Nielsen.)
Jeannette Woerner
Technical University of Dortmund, Germany
A new view on fractional Lévy motions
Some recent results on financial data, both for high and low frequency, indicate that classical semimartingales might not be sufficient to capture all empirical characteristics, such as correlation structure and path regularity. Models based on fractional Brownian motion are able to reproduce these features, but lead to normally distributed increments. As an alternative leading to increments distributed as an arbitrary infinitely divisible distribution we propose a new class of fractional Lévy motions, which can be described by two parameters, the
Hurst parameter and the Blumenthal-Getoor index of the underlying Lévy process. We deduce the important properties of this class of processes and show that they are also suitable for reproducing features such as the spikes in electricity data. (Based on joint work with Sebastian Engelke).